Apparent bug in 4.2 version DSolve ?
- To: mathgroup at smc.vnet.net
- Subject: [mg40688] Apparent bug in 4.2 version DSolve ?
- From: jimd at linfield.edu (Jim Diamond)
- Date: Sun, 13 Apr 2003 02:19:39 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
I have been trying to use a notebook I generated under an earier
version of Mathematica (3.0.0.0), and I have run into a difficulty:
for the differential equation of interest
y''[z,v] + (2 v + 1 - z*z) y[z,v] ==0
the routine DSolve produces two solutions
(a) y1[z,v] = E^(-z^2 /2) HermiteH[v,z]
and
(b) y2[z,v]= E^(-z^2 /2) Hypergeometric1F1[-v/2,1/2,z^2]
The Wronksian of these two solution is
W = 2^v Sqrt[Pi] v / Gamma[1 - v/2]
which vanishes when v is an even positive integer, so these
two solutions are in fact linearly dependent when v is an even
positive integer.
And of course the Hermite polynomials HermiteH[v,z] are even functions
of z when v is an even integer, so it is clear that the general
solution prouced by DSolve for an arbitrary parameter v does not
include the solution of odd parity
(c) y3[z,v] = E^(-z^2 /2) Hypergeometric1F1[(1-v)/2,3/2,z^2]
The Wronskian of (b) and (c) is
W = 1
When one solves the differential equation
y''[z] + (2 (2 k) + 1 - z*z) y[z] ==0
where k is an integer,
then one gets the two linearly independent solutions (b) and (c).
But isn't the whole point of having an analytic general solution
avoiding having to obtain explicit solutions to a differential
equation each time the parameters change?
I wrote to Wolfram about this issue last week.
Hardware: AMD Athlon Processor
1,572,340 kB RAM
OS: Windows 2000 5.00.2195 Service Pack 3
also Windows 98 SE 4.10.2222 A
Mathematica: 4.2.1.0 (Windows)
I am troubled by this error.
Sincerely,
Jim
--
Jim Diamond
Linfield College Chemistry Department
McMinnville, OR 97128
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